I want to evaluate an expression of the form $\nabla_\lambda (Y^\eta \partial_\eta Z^\rho)$.$$\nabla_\lambda (Y^\eta \partial_\eta Z^\rho).\tag{1}$$
I'm not sure about how to write down the connection term. My guess is:
$$\nabla_\lambda (Y^\eta \partial_\eta Z^\rho) = \partial_\lambda (Y^\eta \partial_\eta Z^\rho) + Y^\eta \partial_\eta (\Gamma^\rho_{\lambda \alpha} Z^\alpha ) $$$$\nabla_\lambda (Y^\eta \partial_\eta Z^\rho) = \partial_\lambda (Y^\eta \partial_\eta Z^\rho) + Y^\eta \partial_\eta (\Gamma^\rho_{\lambda \alpha} Z^\alpha ) \tag{2}$$
Is this correct?
